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The Tenth British Mathematical Olympiad
1974年第十届英国奥林匹克数学竞赛
  1. C is the curve y = 4x2/3 for x ≥ 0 and C' is the curve y = 3x2/8 for x ≥ 0. Find curve C" which lies between them such that for each point P on C" the area bounded by C, C" and a horizontal line through P equals the area bounded by C", C and a vertical line through P .
  2. S is the set of all 15 dominoes (m, n) with 1 ≤ m ≤ n ≤ 5. Each domino (m, n) may be reversed to (n, m). How many ways can S be partitioned into three sets of 5 dominoes, so that the dominoes in each set can be arranged in a closed chain: (a, b), (b, c), (c, d), (d, e), (e, a) ?
  3. Show that there is no convex polyhedron with all faces hexagons .
  4. A is the 16 x 16 matrix (ai,j). a1,1 = a2,2 = ... = a16,16 = a16,1 = a16,2 = ... = a16,15 = 1 and all other entries are 1/2. Find A-1 .
  5. In a standard pack of cards every card is different and there are 13 cards in each of 4 suits. If the cards are divided randomly between 4 players, so that each gets 13 cards, what is the probability that each player gets cards of only one suit ?
  6. ABC is a triangle. P is equidistant from the lines CA and BC. The feet of the perpendiculars from P to CA and BC are at X and Y. The perpendicular from P to the line AB meets the line XY at Z. Show that the line CZ passes through the midpoint of AB.
  7. b and c are non-zero. x3 = bx + c has real roots α, β, γ. Find a condition which ensures that there are real p, q, r such that β = pα2 + qα + r, γ = pβ2 qβ+ r, α = pγ2 + qγ + r.
  8. p is an odd prime. The product (x + 1)(x + 2) ... (x + p - 1) is expanded to give ap-1xp-1 + ... + a1x + a0. Show that ap-1 = 1, ap-2 = p(p-1)/2!, 2ap-3 = p(p-1)(p-2)/3! + ap-2(p-1)(p-2)/2!, ... , (p-2)a1 = p + ap-2(p-1) + ap-3(p-2) + ... + 3a2, (p-1)a0 = 1 + ap-2 + ... + a1. Show that a1, a2, ... , ap-2 are divisible by p and (a0 + 1) is divisible by p. Show that for any integer x, (x+1)(x+2) ... (x+p-1) - xp-1 + 1 is divisible by p. Deduce Wilson's theorem that p divides (p-1)! + 1 and Fermat's theorem that p divides xp-1 - 1 for x not a multiple of p.
  9. A uniform rod is attached by a frictionless joint to a horizontal table. At time zero it is almost vertical and starts to fall. How long does it take to reach the table? You may assume that ∫ cosec x dx = log |tan x/2|.
  10. A long solid right circular cone has uniform density, semi-vertical angle x and vertex V. All points except those whose distance from V lie in the range a to b are removed. The resulting solid has mass M. Show that the gravitational attraction of the solid on a point of unit mass at V is 3/2 GM(1 + cos x)/(a2 + ab + b2).
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